Let X1,..., X, be an iid random sample from the Binomial(k, p) distribution, where p is known butk is unknown. For example, this could happen in an experiment where we flip a coin that we knowis fair and observe x; heads, but we do not know how many times the coin was flipped. Supposewe observe values x1,..., xn of our random sample. In this problem, you will show that there is aunique maximum likelihood estimator of k.(a)What is the likelihood function, L(k)? Why is maximization by differentiationdifficult here?(b)of k. What is the value of the likelihood when k < ¸max x;? Based on this, what can youWe have to take a different approach to finding the maximum likelihood estimatori=1,...,nconclude about the range of values the maximum likelihood estimator can take?(c)at k and k +1,Based on part (a), what can you conclude about the ratio of two likelihoods evaluatedL(k)L(k – 1)'L(k + 1)L(k)|for k in the range found in part (b)?(d)Using part (c), show that the maximum likelihood estimator is the value of k thatsatisfiesnnII (1 - ) < (1 – p)" < II(!Xik +1i=1i=for k > max;=1,...,n i.(e)part (d). This description of the MLE for k was found by Feldman and Fox (1968).Show that there is only one unique integer value of k that satisfies the equation in

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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7. Workers at a large toxic ceanup project are concerned that their white blood cell counts may have been reduced. Let x be a random variable that represents white blood cell count per cubic millimeter of whole blood in a healthy adult. Then u-7500 and a -1750 (Reference: Diagnostic Tests with Nursing Applications, S. Loeb). A random sample of n =50 workers from the toxic cleanup site were given a blood test that showed x = 6820. What is the probability that, for healthy adults, I will be this low or lower?
2. City officials are interested in identifying the most dangerous intersections in State College. The city records the number of car accidents at each of n intersections over the course of the year. Let's assume that the number of accidents for a particular intersection i is denoted by X₁. We are provided with a simple random sample of accidents at n intersections, X₁,..., Xn. Let's assume the following model, i.i.d. Xi Pois (X) where A is an unknown parameter of interest (i.e., the unknown population accident rate). (a) the sufficient statistic? Show that the MLE of λ is X. Find its sufficient statistic. Is the MLE a function of Find the bias, variance, and MSE of ÂMLE. Does the MLE achieve the Cramér-Rao's lower bound? (b) (c) (d) Find the asymptotic distribution of the MLE for A. (e) Now, the officials are interested in the total number of accidents expected in those n intersections, that is, E(X1 X₂) = nλ. Let us denote this expected total as (= nλ). Find the MLE of 0. (Hint:…
Group A is a sample of 25 people and has a Sum of Squares of 800 Group B is a sample of 20 people and has a Sum of Squares of 1600 Use the information from both samples to estimate the variance of the null hypothesis distribution?
1. For each random variable described below, give its distribution (which will be Binomial, Geometric, Hypergeometric, Pascal, or Poisson) and the parameters of that distribution. (a) You are working on a difficult homework assignment with 5 problems. Every minute, you have an idea for the problem you are working on, but it only has a 5% chance of working. If it doesn’t work, you keep working on that problem, and if it works, you move on to the next problem. The random variable W is the time (in minutes) that it takes you to finish the homework assignment. (b) A standard 52-card deck is split equally between two players. The random variable X is the number of aces received by the first player. (There are four aces in the deck.) Note: this is a key parameter when playing the “card game” “War”. (c) Every time you turn on the light, the light bulb has a 1% chance of burning out and you have to replace it. The random variable Y is the number of times you turn on the light before you have…
2. Suppose X is a binomial random variable with n = 5 and p = 0.5. a. Display?(?) in tabular form. b. Compute the mean and the variance of x.
17. Conditional Distributions Consider two random variables X and Y for which we know Pr(X = x) = Pr(X = x|Y = y)for all possible values of x and y. In this case we would say that X and Y are.... Pick ONE option Independent Perfectly correlated Equal to each other More than one of these choices None of these choices
Suppose X and Y are two independent random variables with: X Y mean 2 -4 st dev 3 3 The mean of X+Y is .The standard deviation of X+Y is (to two places after the decimal).
3. An integer is randomly chosen from the set 1, 2,..., 100. If this integer is divisible by 2 we let Y = 0, and if it is not divisible by 2 but is divisible by 3 then Y = 1. We let Y = 2 in all other cases. Find mean and variance of Y.
(Don't write handwriting. .)
1. If a random variable X takes the value 1,2,3,4 suchthat 2P(X = 1) = 3P(X = 2) = P(X = 3) = 5P(X = 4).Find the probability distribution of X.
1. Six batteries taken at random from a large lot were subjected to life tests with the following results: 19, 22, 18, 16, 25, and 20 hours. Find the probability that the sample mean of x= 20 hours, as observed here, or one less than 20 hours could arise if the true mean of the lot of batteries is 21.5 hours, assuming a known variance of o = 10 hours, for the individual batteries in the lot. %3D
Each scenario shows the data used for a randomisation test on a difference in two proportions. Both have a sample size of n = 30, with half being assigned to Treatment A and half to Treatment B. Scenario 1 has a difference in proportions of 0.4. Scenario 2 has a difference in proportions of 0.2. If we ran a randomisation test for each scenario, the one that would result in a smaller tail proportion in the resulting re-randomisation distribution is: Scenario 2 Scenario 1 Data Yes Yes Data Yes Yes 01 4 LO 5 02 7 No 03 No 0.2 11 →No 05 11 0.4 06 0.7 10 67 8 No DO 4 09 B B 10

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